Does a wave have accelaration ( wave in the sense the disturbance)?

The wave itself does not have an acceleration. It will have a fixed velocity ( depending on the medium that is vibrating. ) But the elements that make up the wave will have an instantaneous acceleration - usually the same as SHM acceleration. So you have to be careful about what you are describing. You can find out more about the distinction between the two circumstances by looking up "group velocity" which deals with the wave as a whole, and "phase velocity " which deals with the speeds and accelerations of the elements inside the wave envelope.